Popular Existing Methods: Even though the best subset selection problem is classical, for many years, it is generally believed that the problem is not feasible for large values of \(p\) . Relatively recent work [1]} formulates (REF ) as a mixed integer optimization (MIO) problem and demonstrates that the best subset selection problem can be solved for much larger \(p\)  using modern MIO solvers such as in the commercial software Gurobi [2]}, which is not freely available (except for an initial short period). In particular, the best subset selection problem is reformulated as the following MIO problem, containing both continuous constraints and discrete constraints:
\(\begin{aligned}&\operatornamewithlimits{minimize\,}_{\beta ,\, \mathbf {z} \in \mathbb {R}^p } \,\, \frac{1}{n}\Vert  \mathbf {y} - X \beta \Vert _2^2 \\&\operatornamewithlimits{subject\,\, to\,}-b\, z_j \le \beta _j \le b\, z_j,\quad j = 1, \dots , p,\\&\hspace{54.06006pt} z_j \in \lbrace 0, 1\rbrace , \quad j = 1, \dots , p,\\&\hspace{54.06006pt} \sum _{j = 1}^p z_j = k,\end{aligned}\) 
